Researchers have widely used exploratory factor analysis (EFA) to learn the latent structure underlying multivariate data. Rotation and regularised estimation are two classes of methods in EFA that they often use to find interpretable loading matrices. In this paper we propose a new family of oblique rotations based on component-wise $L^p$ loss functions $(0 < p\leq 1)$ that is closely related to an $L^p$ regularised estimator. We develop model selection and post-selection inference procedures based on the proposed rotation method. When the true loading matrix is sparse, the proposed method tends to outperform traditional rotation and regularised estimation methods in terms of statistical accuracy and computational cost. Since the proposed loss functions are nonsmooth, we develop an iteratively reweighted gradient projection algorithm for solving the optimisation problem. We also develop theoretical results that establish the statistical consistency of the estimation, model selection, and post-selection inference. We evaluate the proposed method and compare it with regularised estimation and traditional rotation methods via simulation studies. We further illustrate it using an application to the Big Five personality assessment.

翻译：研究者广泛使用探索性因子分析（EFA）来学习多变量数据背后的潜在结构。旋转和正则化估计是EFA中常用于寻找可解释载荷矩阵的两类方法。本文提出一类新的斜交旋转方法，其基于分量方向的$L^p$损失函数$(0 < p \leq 1)$，与$L^p$正则化估计量密切相关。我们基于所提出的旋转方法开发了模型选择和后选择推断程序。当真实载荷矩阵稀疏时，所提方法在统计精度和计算成本方面往往优于传统旋转和正则化估计方法。由于所提出的损失函数非光滑，我们开发了一种迭代重加权梯度投影算法来求解优化问题。我们还建立了理论结果，证明了估计、模型选择和后选择推断的统计相合性。通过模拟研究评估所提方法，并将其与正则化估计和传统旋转方法进行比较。我们进一步通过"大五人格"评估的应用实例加以说明。